Consider the problem of trying to identify which $n$-dimensional compact complex manifolds can be endowed with a Kähler metric.

$\underline{n = 1}:$ Any hermitian metric on a Riemann surface is a Kähler metric as the Kähler form is a two-form, and all two-forms are closed. (This is also true of non-compact Riemann surfaces).

$\underline{n = 2}:$ A necessary and sufficient condition for a complex surface to admit a Kähler metric is that its odd Betti numbers are even.

The condition that odd Betti numbers are even is necessary for a complex manifold to admit a Kähler metric, but not sufficient for $n \geq 3$. To see it is not sufficient, consider the example of Hironaka which gives a deformation of three-dimensional Kähler manifolds to a non-Kähler manifold. The Betti numbers are invariant under diffeomorphisms, so the central fibre of the deformation has odd Betti numbers even (as it is diffeomorphic to a Kähler manifold) but it is not itself Kähler.

In general, for $n \geq 3$, it is not easy to determine when a compact complex manifold can be equipped with a Kähler metric. In the simplest of these cases, $n = 3$, are we any closer to solving this problem?

What conditions (necessary or sufficient) are there for a three-dimensional compact complex manifold to admit a Kähler metric? How far are we from a single necessary and sufficient condition (like we have for $n = 1$ and $2$)?

Of course I'm interested in results which apply for all $n$, but I'm guessing that there are some results specifically for $n = 3$.